This is not how the iPad version works.
The Edit button only allows you to Delete or Change the position of entries.
Connection Settings is the button which is at the right under the connect button.
When you go into Connection Settings for a Manual Connection, then you can change the Nickname.
When you go into Connection Settings for a Discovered Connection, then the Nickname is greyed out.
I am using version 2.2.223 of PocketCloud Pro which is the latest release.
I shouldn't have to explain how the software works to you, Josh. If I'm missing something, then I'm sorry, but I don't think so.
It seems that maybe this isn't possible at this time, if so, please make a request to the development team. Changing the Nickname within the PocketCloud Companion would be best.